This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212343 #60 Apr 03 2022 08:30:14
%S A212343 0,0,5,18,42,80,135,210,308,432,585,770,990,1248,1547,1890,2280,2720,
%T A212343 3213,3762,4370,5040,5775,6578,7452,8400,9425,10530,11718,12992,14355,
%U A212343 15810,17360,19008,20757,22610,24570,26640,28823,31122,33540,36080,38745,41538,44462,47520,50715,54050,57528
%N A212343 a(n) = (n+1)*(n-2)*(n-3)/2.
%C A212343 Sequence of coefficients of x^1 in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%C A212343 Is this row 2 of the convolution array A213819? - _Clark Kimberling_, Jul 04 2012
%H A212343 Colin Barker, <a href="/A212343/b212343.txt">Table of n, a(n) for n = 2..1000</a>
%H A212343 S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243 [math.CO], 2012.
%H A212343 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A212343 For n>=4, a(n) = (n-3)*A212342(n-1).
%F A212343 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>7. - _Colin Barker_, Jul 10 2015
%F A212343 G.f.: -x^4*(2*x-5) / (x-1)^4. - _Colin Barker_, Jul 10 2015
%F A212343 From _Amiram Eldar_, Apr 03 2022: (Start)
%F A212343 Sum_{n>=4} 1/a(n) = 23/72.
%F A212343 Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/3 - 55/72. (End)
%t A212343 QQ0[t, x] = (1 - (1-4*x*t)^(1/2)) / (2*x*t); QQ1[t, x] = 1/(1 - t*QQ0[t, x]); QQ2[t, x] = (1 + t*(QQ1[t, x] - QQ0[t, x]))/(1 - t*QQ0[t, x]); QQ3[t, x] = (1 + t*(QQ2[t, x] - QQ0[t, x] + t*(QQ1[t, x] - QQ0[t, x])))/(1 - t*QQ0[t, x]); CoefficientList[Coefficient[Simplify[Series[QQ3[t, x], {t, 0, 35}]],x],t] (* _Robert Price_, Jun 04 2012 *)
%t A212343 LinearRecurrence[{4,-6,4,-1},{0,0,5,18},60] (* _Harvey P. Dale_, Mar 15 2018 *)
%o A212343 (PARI) Vec(-x^4*(2*x-5)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Jul 10 2015
%Y A212343 Partial sums are in A241765.
%Y A212343 Cf. similar sequences of the type m*(m+1)*(m+k)/2 listed in A267370.
%Y A212343 Cf. also A212342.
%K A212343 nonn,easy
%O A212343 2,3
%A A212343 _N. J. A. Sloane_, May 09 2012
%E A212343 a(10)-a(35) from _Robert Price_, Jun 02 2012
%E A212343 Entry revised by _N. J. A. Sloane_, Sep 10 2016